Exponential Stability of Nonlinear Systems With Delayed Impulses and Applications

• Published in: IEEE transactions on automatic control Vol. 64; no. 10; pp. 4024 - 4034
• Main Authors: Li, Xiaodi, Song, Shiji, Wu, Jianhong
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Average dwell time (ADT); Cellular communication; Chaos theory; Control systems; Control theory; Delay effects; delayed impulse; Dwell time; Dynamic stability; Impulses; impulsive system; Linear matrix inequalities; Mathematical analysis; Matrix methods; Neural networks; Nonlinear systems; Numerical stability; stability; Stability criteria; Synchronism; Synchronization; System dynamics; Time lag
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2019.2905271

We consider a class of nonlinear impulsive systems with delayed impulses, where the time delays in impulses exist between two consecutive impulse instants. Based on the impulsive control theory and the ideas of average dwell time (ADT), a set of Lyapunov-based sufficient conditions for globally exponential stability are obtained. It is shown that the time delay in impulses exhibits double effects (i.e., negative or positive effects) on system dynamics, namely, it may destroy the stability and lead to undesired performance, and conversely, it may stabilize an unstable system and achieve better performance. Then, we apply the theoretical results to synchronization control of chaotic systems and design some impulsive controllers that are formalized in terms of linear matrix inequality and ADT-like conditions. Two numerical examples including chaotic cellular neural network and Chua's oscillator are given to illustrate the applicability of the presented impulsive control schemes. Our examples show that, under some conditions, the impulsive synchronization of chaotic systems can be achieved via the input delays in impulses.